Question

The set of all function from natural numbers to natural numbers -- countable ?

Answer 1

Every real number in (0,1) has either one or two decimal representations. (Some have two, like 0.1000… which is equal to 0.0999….) But any way, each real number in (0,1) gives at least one decimal representation. And that decimal representation can be viewed as a function from N to N by taking the nth digit after the decimal.